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The energetically independent linear wave and geostrophic (vortex) solutions are shown to be a complete basis for velocity and density variables $(u,v,w,\rho )$ in a rotating non-hydrostatic Boussinesq fluid with arbitrary stratification and non-periodic vertical boundaries. This work extends the familiar wave-vortex decomposition for triply periodic domains with constant stratification. As a consequence of the decomposition, the fluid can be unambiguously separated into decoupled linear wave and geostrophic components at each instant in time, without the need for temporal filtering. The fluid can then be diagnosed for temporal changes in wave and geostrophic coefficients at each unique wavenumber and mode, including those that inevitably occur due to nonlinear interactions. We demonstrate that this methodology can be used to determine which physical interactions cause the transfer of energy between modes by projecting the nonlinear equations of motion onto the wave-vortex basis. In the particular example given, we show that an eddy in geostrophic balance superimposed with inertial oscillations at the surface transfers energy from the inertial oscillations to internal gravity wave modes. This approach can be applied more generally to determine which mechanisms are involved in energy transfers between wave and vortices, including their respective scales. Finally, we show that the nonlinear equations of motion expressed in a wave-vortex basis are computationally efficient for certain problems. In cases where stratification profiles vary strongly with depth, this approach may be an attractive alternative to traditional spectral models for rotating Boussinesq flow. 

The unsteady Ekman problem involves finding the response of the near-surface currents to wind stress forcing under linear dynamics. Its solution can be conveniently framed in the frequency domain in terms of a quantity that is known as the transfer function, the Fourier transform of the impulse response function. In this paper, a theoretical investigation of a fairly general transfer function form is undertaken with the goal of paving the way for future observational studies. Building on earlier work, we consider in detail the transfer function arising from a linearly-varying profile of the vertical eddy viscosity, subject to a no-slip lower boundary condition at a finite depth. The horizontal momentum equations, rendered linear by the assumption of horizontally uniform motion, are shown to transform to a modified Bessel’s equation for the transfer function. Two self-similarities, or rescalings that each effectively eliminate one independent variable, are identified, enabling the dependence of the transfer function on its parameters to be more readily assessed. A systematic investigation of asymptotic behaviors of the transfer function is then undertaken, yielding expressions appropriate for eighteen different regimes, and unifying the results from numerous earlier studies. A solution to a numerical overflow problem that arises in the computation of the transfer function is also found. All numerical code associated with this paper is distributed freely for use by the community. 

This dataset uses trajectory data from a large set of drifters to extract and analyze displacement signals associated with coherent eddies in the Gulf of Mexico, using a multivariate wavelet ridge analysis as presented in Lilly and Pérez-Brunius (2021). The data includes eddy displacement signals for all ridges, as well as the time-varying ellipse parameters and estimated ellipse center location. The instantaneous frequency is also included, as is the instantaneous bias estimate derived by Lilly and Olhede (2012). The data are organized as appended trajectory data that can be readily separated through the use of the "ids" field.  The ridge length (\(L\)),and ridge-averaged circularity (\(\overline{\xi}\))  are also included, as is measure of statistical significance denoted by (\(\rho\)). The dataset is available for download as a NetCDF file.

Lilly, J. M. and P. Pérez-Brunius (2021).  Extracting statistically significant eddy signals from large Lagrangian datasets using wavelet ridge analysis, with application to the Gulf of Mexico. Nonlinear Processes in Geophysics, 28: 181–212. https://doi.org/10.5194/npg-28-181-2021. 

Lilly, J. M. and Olhede, S. C.: Analysis of modulated multivariate oscillations, IEEE T. Signal Proces., 60, 600–612, 2012. 10.1109/TSP.2011.2173681

The GOMED database is a product of the Gulf of Mexico Research Consortium (CIGoM) and was partially funded by the CONACYT-SENER-Hydrocarbons Sector Fund, Mexico, project 201441.See database webpage with additional information, as well as request for download form (https://giola.cicese.mx/database/GOMED) {"references": ["Lilly, J. M. and Olhede, S. C.: Higher-order properties of analytic wavelets, IEEE T. Signal Proces., 57, 146\u2013160,\u00a0https://doi.org/10.1109/TSP.2008.2007607, 2009.", "Lilly, J. M. and Olhede, S. C.: Analysis of modulated multivariate oscillations, IEEE T. Signal Proces., 60, 600\u2013612, 2012."]} 

Abstract. A method for objectively extracting the displacement signals associated with coherent eddies from Lagrangian trajectories is presented, refined, and applied to a large dataset of 3770 surface drifters from the Gulf of Mexico. The method, wavelet ridge analysis, is a general method for the analysis of modulated oscillations, here modified to be more suitable to the eddy-detection problem. A means for formally assessing statistical significance is introduced, addressing the issue of false positives arising by chance from an unstructured turbulent background and opening the door to confident application of the method to very large datasets. Significance is measured through a frequency-dependent comparison with a stochastic dataset having statistical and spectral properties that match the original, but lacking organized oscillations due to eddies or waves. The application to the Gulf of Mexico reveals major asymmetries between cyclones and anticyclones, with anticyclones dominating at radii larger than about 50 km, but an unexpectedly rich population of highly nonlinear cyclones dominating at smaller radii. Both the method and the Gulf of Mexico eddy dataset are made freely available to the community for noncommercial use in future research. 

Drifters deployed in close proximity collectively provide a unique observational data set with which to separate mesoscale and submesoscale flows. In this paper we provide a principled approach for doing so by fitting observed velocities to a local Taylor expansion of the velocity flow field. We demonstrate how to estimate mesoscale and submesoscale quantities that evolve slowly over time, as well as their associated statistical uncertainty. We show that in practice the mesoscale component of our model can explain much first and second-moment variability in drifter velocities, especially at low frequencies. This results in much lower and more meaningful measures of submesoscale diffusivity, which would otherwise be contaminated by unresolved mesoscale flow. We quantify these effects theoretically via computing Lagrangian frequency spectra, and demonstrate the usefulness of our methodology through simulations as well as with real observations from the LatMix deployment of drifters. The outcome of this method is a full Lagrangian decomposition of each drifter trajectory into three components that represent the background, mesoscale, and submesoscale flow. 

Abstract Observations from the past decades have promoted the idea of a long-lived anticyclonic vortex residing in the Lofoten Basin. Despite repeatedly recorded intense anticyclones, the observations cannot firmly decide whether the signature is of a single vortex or a succession of ephemeral vortices. A vortex persisting for decades requires some reinvigoration mechanism. Wintertime convection and vortex merging have been proposed candidates. We examine Lofoten Basin vortex dynamics using a high-resolution regional ocean model. The model is initialized from a coarser state with a weak eddy field. The slope current intensifies and sheds anticyclonic eddies that drift into the basin. After half a year, an anticyclone arrives at the center, providing the nucleus for a vortex that remains distinct throughout the simulation. Analyses show that this vortex is regenerated by repeated absorption and vertical stacking of lighter anticyclones. This compresses and—in concert with potential vorticity conservation—intensifies the combined vortex, which becomes more vertically stratified and also expels some fluid in the process. Wintertime convection serves mainly to vertically homogenize and densify the vortex, rather than intensifying it. Further, topographic guiding of anticyclones shed from the continental slope is vital for the existence and reinvigoration of the Lofoten vortex. These results offer a new perspective on the regeneration of oceanic anticyclones. In this scenario the Lofoten vortex is maintained through repeated merging events. Fluid remains gradually exchanged, although the vortex is identifiable as a persistent extremum in potential vorticity. 

A comprehensive method is provided for smoothing noisy, irregularly sampled data with non-Gaussian noise using smoothing splines. We demonstrate how the spline order and tension parameter can be chosen a priori from physical reasoning. We also show how to allow for non-Gaussian noise and outliers that are typical in global positioning system (GPS) signals. We demonstrate the effectiveness of our methods on GPS trajectory data obtained from oceanographic floating instruments known as drifters. 

Abstract The Lofoten Basin is the largest oceanic reservoir of heat in the Nordic Seas, and the site of important heat fluxes to the atmosphere. An intense permanent anticyclone in the basin impacts the regional hydrography, energetics, and ecosystem. Repeated sampling of this Lofoten Basin Eddy from dedicated cruises, autonomous profiling gliders, and acoustically-tracked subsurface floats enables the documentation of its dynamics and energetics over the course of 15 months. The eddy core, in nearly solid-body rotation, exhibits an unusually low vertical vorticity close to the local inertial frequency and important strain rates at the periphery. Subsurface floats as deep as 800 m are trapped within the core for their entire deployment duration (up to 15 months). The potential vorticity is reduced in the core by two orders of magnitude relative to the surroundings, creating a barrier. In the winter, this barrier weakens and lateral exchanges and heat flux between the eddy and the surroundings increase, apparently the result of dynamical instabilities and a possible eddy merger. Based on a simple energy budget, the dissipation timescale for the eddy energy is three years, during which wintertime convection seasonally modulates potential and kinetic energy.